The constraint satisfaction problem, parameterized by a relational structure, provides a general framework for expressing computational decision problems. Already the restriction to the class of all finite structures forms an interesting microcosm on its own, but to express decision problems in temporal reasoning one has to take a step beyond the finite-domain realm. An important class of templates used in this context are temporal structures, i.e., structures over ℚ whose relations are first-order definable using the usual countable dense linear order without endpoints. In the standard setting, which allows only existential quantification over input variables, the complexity of finite and temporal constraints has been fully classified. In the quantified setting, i.e., when one also allows universal quantifiers, there is only a handful of partial classification results and many concrete cases of unknown complexity. This paper presents a significant progress towards understanding the complexity of the quantified constraint satisfaction problem for temporal structures. We provide a complexity dichotomy for quantified constraints over the Ord-Horn fragment, which played an important role in understanding the complexity of constraints both over temporal structures and in Allen’s interval algebra. We show that all problems under consideration are in P or coNP-hard. In particular, we determine the complexity of the quantified constraint satisfaction problem for (ℚ;x = y⇒ x ≥ z), hereby settling a question open for more than ten years.

Abstract. We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities [Formula: see text]) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain constraint satisfaction problems (CSPs) studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of Monotone Monadic SNP (MMSNP) sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al. In particular, the bounded width hierarchy collapses in those cases as well. Our results extend the scope of theorems of Barto and Kozik characterizing bounded width for finite structures and show the applicability of infinite-domain CSPs to other fields.

We prove that QCSP(ℕ; x = y → y = z) is PSpace-complete, settling a question open for more than ten years. This completes the complexity classification for the QCSP over equality languages as a trichotomy between Logspace, NP-complete and PSpace-complete. We additionally settle the classification for bounded alternation QCSP(Γ), for Γ an equality language. Such problems are either in Logspace, NP-complete, co-NP-complete or rise in complexity in the Polynomial Hierarchy.

We prove that relational structures admitting specific polymorphisms (namely, canonical pseudo-WNU operations of all arities n ≥ 3) have low relational width. This implies a collapse of the bounded width hierarchy for numerous classes of infinite-domain CSPs studied in the literature. Moreover, we obtain a characterization of bounded width for first-order reducts of unary structures and a characterization of MMSNP sentences that are equivalent to a Datalog program, answering a question posed by Bienvenu et al.. In particular, the bounded width hierarchy collapses in those cases as well.

The relational width of a finite structure, if bounded, is always (1, 1) or (2, 3). In this paper we study the relational width of first-order expansions of finitely bounded homogeneous binary cores where binary cores are structures with equality and some anti-reflexive binary relations such that for any two different elements a, b in the domain there is exactly one binary relation R with (a, b) ϵ R.

Solving the algebraic dichotomy conjecture for constraint satisfaction problems over structures first-order definable in countably infinite finitely bounded homogeneous structures requires understanding the applicability of local-consistency methods in this setting. We study the amount of consistency (measured by relational width) needed to solve CSP(?) for first-order expansions ? of countably infinite homogeneous graphs ℋ := (A; E), which happen all to be finitely bounded. We study our problem for structures ? that additionally have bounded strict width, i.e., for which establishing local consistency of an instance of CSP(?) not only decides if there is a solution but also ensures that every solution may be obtained from a locally consistent instance by greedily assigning values to variables, without backtracking. Our main result is that the structures ? under consideration have relational width exactly (2, ?_ℋ) where ?_ℋ is the maximal size of a forbidden subgraph of ℋ, but not smaller than 3. It beats the upper bound: (2 m, 3 m) where m = max(arity(?)+1, ?, 3) and arity(?) is the largest arity of a relation in ?, which follows from a sufficient condition implying bounded relational width given in [Manuel Bodirsky and Antoine Mottet, 2018]. Since ?_ℋ may be arbitrarily large, our result contrasts the collapse of the relational bounded width hierarchy for finite structures ?, whose relational width, if finite, is always at most (2,3).

We study the complexity of the inference problem for propositional circumscription (the minimal inference problem) over arbitrary finite domains. The problem is of fundamental importance in nonmonotonic logics and commonsense reasoning. The complexity of the problem for the two-element domain has been completely classified. In this article, we classify the complexity of the problem over all conservative languages. We consider a version of the problem parameterized by a set of relations (a constraint language), from which we are allowed to build a knowledge base, and where a linear order used to compare models is a part of an input. We show that in this setting the problem is either Π P 2 -complete, coNP-complete, or in P. The classification is based on a coNP-hardness proof for a new class of languages, an analysis of languages that do not express any member of the class, and a new general polynomial-time algorithm solving the minimal inference problem for a large class of languages.

Abduction is a form of nonmonotonic reasoning that looks for an explanation, built from a given set of hypotheses, for an observed manifestation according to some knowledge base. Following the concept behind the Schaefer's parametrization CSP(Gamma) of the Constraint Satisfaction Problem (CSP), we study here the complexity of the abduction problem Abduction(Gamma, Hyp, M) parametrized by certain (omega-categorical) infinite relational structures Gamma, Hyp, and M from which a knowledge base, hypotheses and a manifestation are built, respectively. We say that Gamma has local-to-global consistency if there is k such that establishing strong k-consistency on an instance of CSP(Gamma) yields a globally consistent (whose every solution may be obtained straightforwardly from partial solutions) set of constraints. In this case CSP(Gamma) is solvable in polynomial time. Our main contribution is an algorithm that under some natural conditions decides Abduction(Gamma, Hyp, M) in P when Gamma has local-to-global consistency. As we show in the number of examples, our approach offers an opportunity to consider abduction in the context of spatial and temporal reasoning (qualitative calculi such as Allen's interval algebra or RCC-5) and that our procedure solves some related abduction problems in polynomial time.

Abduction is a form of nonmonotonic reasoning that looks for an explanation for an observed manifestation according to some knowledge base. One form of the abduction problem studied in the literature is the propositional abduction problem parameterized by a structure \Gamma over the two-element domain. In that case, the knowledge base is a set of constraints over \Gamma, the manifestation and explanation are propositional formulas. In this paper, we follow a similar route. Yet, we consider abduction over infinite domain. We study the equality abduction problem parameterized by a relational first-order structure \Gamma over the natural numbers such that every relation in \Gamma is definable by a Boolean combination of equalities, a manifestation is a literal of the form (x = y) or (x != y), and an explanation is a set of such literals. Our main contribution is a complete complexity characterization of the equality abduction problem. We prove that depending on \Gamma, it is \Sigma^P_2-complete, or NP-complete, or in P.

The first-order theory of dense linear orders without endpoints is well-known to be PSPACE-complete. We present polynomial-time tractability results for fragments of this theory which are defined by syntactic restriction, in particular, our fragments can be described using the framework of quantified constraint satisfaction over Ord-Horn clauses.